Question: Two marbles are drawn from a jar with 10 marbles. If all marbles are either red or blue, is the probability that both marbles selected will be red greater than \(\frac{3}{5}\)?

1. The probability that both marbles selected will be blue is less than \(\frac{1}{10}\)
2. Atleast 60% of the marbles in the jar are red

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation
Approach Solution (1):

(1) The probability that both marbles selected will be blue is less than \(\frac{1}{10}\)
This implies that \(\frac{B}{10} * \frac{B-1}{9}<\frac{1}{10}\). So, we have that B (B – 1) < 9, thus B < 4, so the number of red marbles in the jar is 7, 8, 9, or 10
Not sufficient

(2) Atleast 60% of the marbles in the jar are red. This implies that the number of red marbles is 6 or more
Not sufficient

(1) + (2) from above, we have that R > 6
Not sufficient

Correct Option: E

Approach Solution (2):

We have 10 marbles blue and red. We need to check the probability of selecting both red marbles f is greater than 0.6
S1- We have probability of both blue is < \(\frac{1}{10}\)
So,
\(\frac{b}{10} * \frac{b-1}{9} < \frac{1}{10}\)
\(b(b-1) < \frac{90}{10}\)
\(b(b-1) < 9\)

So b can be 2, 3

If b is 2 then the probability of both red marbles is \(\frac{8}{10}*\frac{7}{9}\)which 0.622 hence greater than \(\frac{3}{5}\)but if b is 3 then the probability of both red marbles is \(\frac{7}{10}*\frac{6}{9}\)which is 0.466 so we can’t come down to a single result.
Hence not sufficient

S2: We have 60% of the marbles are red so again we have the same confusion as in S1 when the red marbles are 7 or when they are 8
So not sufficient

Combining also we do not come to any conclusion

Correct Option: E

Approach Solution (3):

We are trying to know whether there are more than 7 red balls, that is 8 or 9 red balls out of the 10 since both of these will make the possibility of two consecutive reds > \(\frac{3}{5}\). (7 reds will give us \(\frac{7}{10} * \frac{6}{9} = \frac{14}{30}\) which is < \(\frac{3}{5}\)… so we need atleast 8).

1. Probability of two consecutive blues is less than \(\frac{1}{10}\)th. For this to be true; there must be either 1 blue, 2 blues, or 3 blues. Based on what we noted earlier, 1 and 2 blues will answer the target question as yes but 3 blues will make the answer no

Insufficient

2. Tell us that there can be 7, 8, or 9 balls. We need to know if there are 8 greater to answer definitely

Insufficient
Combined: We can still have 1 blue and 9 reds in which case target answer is yes or 3 blues and 7 reds in which case target answer is no

Correct Option: E

“Two marbles are drawn from a jar with 10 marbles. If all marbles are either red or blue, is the probability that both marbles selected will be red greater than \(\frac{3}{5}\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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